Dynamics Seminar Spring 2011

The Dynamics seminar will meet on Friday afternoons at 3:30 in DSB C118. All are welcome. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title
14/1/2011 Jason Siefken (UVic) Ergodic Optimization of super-continuous functions
21/1/2011 Michael Schraudner (Univ of Chile) Projective subdynamics of Z^d shifts of finite type
28/1/2011 Antoine Julien (UVic) Complexity for cut and project tilings
4/2/2011    
11/2/2011   No seminar
17/2/2011 Ronnie Pavlov (Denver University) Shifts of finite type with nearly full entropy
25/2/2011   No seminar
4/3/2011 Robert Moody (UVic) On Joan Taylor's hexagonal monotile
11/3/2011 Tom Meyerovitch (UBC) Ergodicity of Poisson Products
18/3/2011 Mike Hochman (Microsoft) Geometric rigidity of times-m invariant measures
25/3/2011 Ayse Sahin (DePaul) The directional entropy of ergodic, zero entropy Z^d actions and the loosely Bernoulli property
31/3/2011 at 1:20 in C130 John Griesmer (UBC) Dense sets of integers via dynamics


Date: 14/1/2011
Speaker: Jason Siefken (UVic)
Title: Ergodic Optimization of super-continuous functions
Abstract: Ergodic optimization is the study of what types of ergodic measures are optimized by certain classes of functional; i.e., for a class of functions \{f_i\}, what are the qualities of measures \mu_i such that \int f_i \mu_i \geq \int f_i \nu for all ergodic measures \nu. The standing conjecture is that "most" functions are optimized by ergodic measures supported on a finite set. In this talk we give a brief history of the results in ergodic optimization as well as introduce the class of Super-continuous functions, where the conjecture holds true.
Date: 21/1/2011
Speaker: Michael Schraudner (Univ of Chile)
Title: Projective subdynamics of Z^d shifts of finite type
Abstract: Motivated by Hochman's notion of subdynamics of a Z^d subshift, we define and examine projective subdynamics of Z^d shifts of finite type (SFTs) where we restrict not only the action but also the phase space. In analogy with the notion of stable and unstable limit sets in cellular automata we distinguish between stable and unstable projective subdynamics. First we review the classification of Z sofic shifts which can (not) appear as projective subdynamics of Z^2 (Z^d) SFTs both in the stable and unstable regime - these are results obtained jointly with Ronnie Pavlov. In a second part of the talk we present results on the projective subdynamics of Z^d SFTs with some uniform mixing condition. In particular there is a compatibility condition assuring the projective subdynamics of Z^d SFTs has to be sofic and if time permits we explain a construction that allows to realize any mixing Z sofic as stable projective subdynamics of some strongly irreducible Z^2 (Z^d) SFT.
Date: 28/1/2011
Speaker: Antoine Julien (UVic)
Title: Complexity for cut and project tilings
Abstract: The study of aperiodic tilings was boosted by the discovery, in the early 1980's, of a strange alloy. While it would deserve to be called a "crystal", its atoms are not arranged following a periodic pattern. The first quasi-crystal was born.

In this talk, I will present the cut and project method, which was used to produce many interesting example, like the Penrose tiling or the Octogonal tiling. The Penrose tiling itself is a model for some quasi-crystals. Tilings produced by this method, while non periodic, are very much "ordered", and are very far from being random tilings.

We measure this "aperiodic order" by using the complexity function, which counts the number of different patterns of a given size. The theorem that I want to present gives the asymptotic order of growth of the complexity function. This result can also be seen as a multi-dimensional generalization of the computation of complexity for cubic billiard sequences.


Date: 4/2/2011
Speaker:
Title:
Abstract:
Date: 17/2/2011
Speaker: Ronnie Pavlov (Denver University)
Title: Shifts of finite type with nearly full entropy
Abstract: Z^d shifts of finite type (or SFTs) are a well-studied class of topological dynamical systems. A Z^d-SFT X is defined by specifying a finite alphabet A and a finite set of forbidden configurations F, and then defining X to be the set of all ways of assigning letters from A to all sites in Z^d such that none of the configurations in F appear. For instance, taking A={0,1} and F={11} defines a Z-SFT consisting of all biinfinite zero-one sequences without consecutive ones.

Z^d-SFTs are generally much more complicated than their one-dimensional counterparts; one illustration of this is given by their measures of maximal entropy. Any topologically mixing Z-SFT has a unique m.m.e., but it was shown by Burton and Steif that not even uniform topological mixing conditions such as strong irreducibility (which often suffice to generalize theorems about Z-SFTs to Z^d) imply that a Z^d-SFT has a unique m.m.e.

We present a new sufficient condition for a Z^d-SFT to have a unique m.m.e., which is expressed purely in terms of topological entropy. Namely, for any d, there is a constant B(d) > 0 so that any nearest-neighbor Z^d-SFT X with alphabet A and entropy at least (log |A|) - B(d) has a unique m.m.e. We will also present some examples and background to illustrate how this result compares to other sufficient conditions in the literature.


Date: 4/3/2011
Speaker: Robert Moody (UVic)
Title: On Joan Taylor's hexagonal monotile
Abstract: In 2010 Joan Taylor announced her discovery of a hexagonal tile that can only tile the plane aperiodically. The tile is a regular hexagon and the tiling is a regular hexagonal tiling, but the tile itself is marked differently on its two sides and the matching rules force the two sides to appear in an aperiodic way. We discuss the family of tilings that arise from the matching rules, which forms a hull and a dynamical system. We also offer a cut and project interpretation through which the tilings can be seen as model sets.  The parity patterns that are created by the Taylor tilings are quite  fascinating and raise many interesting questions. This talk is a report on work in progress with Jeong-Yup Lee at KIAS.
Date: 11/3/2011
Speaker: Tom Meyerovitch (UBC)
Title: Ergodicity of Poisson Products
Abstract: Let $T:X \to X$ be a measure preserving transformation of an infinite-measure space $(X,\mathcal{B},\mu)$ with $\mu(X)=\infty$. Associated with $T$ is a natural probablity-preserving map $T_*$ which acts on discrete countable subsets of $X$, with resepect to the probablity measure defined by Poisson processes on $X$. This map is called the Poisson suspension of $T$. I will review some basic properties of Poisson suspensions. Under the assumption that the transofrmation $T$ is recurrent and ergodic, I will prove ergodicty of the map $T \times T_*$, and discuss applications.
Date: 18/3/2011
Speaker: Mike Hochman (Microsoft/Hebrew University)
Title: Geometric rigidity of times-m invariant measures
Abstract: I will discuss a "local" generalization of Rudolph's measure rigidity theorem for times-2 and times-3 invariant measures. I will explain why times-2 invariant measures of positive entropy cannot be smoothly deformed so as to preserve the measure class, or deformed into times-3 invariant measures. In particular, this shows that in a sense what underlies Rudolph's theorem is not the structure of the abelian action generated by the two maps, but rather properties of the individual maps.
Date: 25/3/2011
Speaker: Ayse Sahin (DePaul)
Title: The directional entropy of ergodic, zero entropy Z^d actions and the loosely Bernoulli property
Abstract: We study the dynamical properties of ergodic, zero entropy Z^d actions as a function of the directional entropy function. The main result relates directional entropy to the orbit equivalence classification of the action. In particular we show how positive directional entropy prevents a zero entropy Z^d action from being loosely Bernoulli, i.e. Kakutani equivalent to an action of rotations. This is joint work with E. A. Robinson.
Date: 31/3/2011
Speaker: John Griesmer (UBC)
Title: Dense sets of integers via dynamics
Abstract: In his ergodic-theoretic proof of Szemerdi's theorem theorem, Furstenberg showed that arithmetic progressions in dense sets of integers can be understood through analysis of special probability measure preserving systems. This strategy was refined by many authors, and we now have the powerful theory of nilfactors, which reduces the study of dense sets of integers to the study of a very specialized class of measure preserving systems. We will outline Furstenberg's strategy and the theory of nilsystems, and apply this to obtain a recent result about sets of the form A+B:={a+b| a in A, b in B} where A is infinite and B has positive upper Banach density.
For previous semesters, see
Spring 2006
Fall 2006
Spring 2007
Spring 2008
Spring 2009
Spring 2010